Independent Samples: Comparing Means
Population 1 mean = μ1Population 2 mean = μ2
Population 1 standard déviation = σ1 Population 2 standard déviation = σ2
Assumptions for a Two Independent Samples Design
We have a simple random sample of observations from a population. We have a simple random sample of observations from a population. The two random samples are independent of each other.
Notation in Two Independent Samples Design
= sample size for first sample (number of observations from Population
= sample size for second sample (number of observations from Population = observed sample mean for the first sample.
= observed sample mean for the second sample.
= observed sample standard deviation for the first sample.
= observed sample standard deviation for the second sample.
Testing the Difference Between Two Means of Independent Samples Design
There are actually two different options for the use of t tests. One option is used whenthe variances of the populations are not equal, and the other option is used when the variancesare equal. To determine whether two sample variances are equal, the researcher
can use an F test.
Note, however, that not all statisticians are in agreement about using the F test before using the t test. Some believe that conducting the F and t tests at the same level of significancewill change the overall level of significance of the t test. Their reasons arebeyond the scope of this course.
Not satisfied
Satisfied
Did not reject Reject
Let’s Do It! 1
Which Version of a Two Independent Samples Test to Use?
Each scenario presents a picture of the distributions of the two populations being compared. Based on these distributions, determine which version of the two-independent samples test to use.
Two Independent Samples Pooled t-Test
We are interested in comparing the population means and , so the parameter of interest is the difference .
Distribution of the Standardized for the Two Independent Samples Scenario when
The quantity
Where , has a t-distribution with degrees of freedom.
Two Independent Samples Pooled t-Test
Assumptions:The first sample is a random sample from a normal population with mean 1. The second sample is a random sample from a normal population with mean 2. The two samples are independent. Normality is less crucial if the sample sizes n1 and n2 are large,
Hypotheses: versus or
versus or
versus .
Data:The two sets of data from which the two sample means and, and the two sample standard deviations and can be computed.
Observed Test Statistic: where
And the t-distribution used hasd.f.= (n1+ n2– 2)
p-value: We find the p-value for the test using the t(n1+ n2 - 2) distribution. The direction of extreme will depend on how the alternative hypothesis is expressed.
EXAMPLE
Comparing Two Headache Treatments
Medical researchers are comparing two treatments for migraine headaches. They wish to perform a double-blind experiment to assess if Treatment B (the new treatment) is significantly better than Treatment A (the standard treatment) using a 5% significance level. Assume equal variances of the populations. The data
(a)State the appropriate hypotheses to be tested. Keep in mind that smaller responses imply a better treatment and Treatment 1 is the new treatment.
vs
(b) Give an estimate of the common (pooled) population standard deviation.
(c) Compute the pooled t-test statistic.
(d) Find the corresponding p-value.
The p-value is the probability of observing a test statistic as large as or larger than the observed value of 1.416, with d.f= 10 +10 -2 =18
Using the TI:
1. Using the tcdf( function.
Using the tcdf( function on the TI we have:
p-value == tcdf(1.416, E99, 18) = 0.0869.
2. Using the 2-SampTTest function under STAT TESTS.
In the TESTS menu located under the STAT button, we select the 4:2-SampTTest option. With the sample means of 22.6 and 19.4, the sample standard deviations of 5.2 and 4.9, and the sample sizes of 10 and 10, we can use the Stats option of this test. The steps and corresponding input and output screens are shown. Notice that you must specify Yes under the Pooled option. The No Pooled option is discussed at the end of this section as another version of our test.
p-value == 0.08688.
(g) State the decision and conclusion using a 5%significance level.
At the 5% significance level we cannot reject the null hypothesis. The samples failed to provide any significant results.
Let’s Do It!
Sample Size / Sample Mean / Sample Standard DeviationDrug 1 / 12 / 5.6 / 1.3
Drug 2 / 14 / 5.0 / 1.8
(a) Assume the two equal population variances and the assumption of independent samples is satisfied. Suppose we can assume each sample is representative of the larger population of potential drug users. One more assumption is required regarding the populations. What is that assumption?
(d) Is the difference between the mean cholesterol reduction for Drug 1 and the mean cholesterol reduction for Drug 2 statistically significant at the 5% level?
Hypothesis:
Test statistic:
P-value:
Decision:
Conclusion:
Homework Page339: 11, 12, 13, 29, 30, 40, 47 (assume variances are equal for all problem)